classification of static and dynamic behavior in chemostat for plasmid ...

Chem. Eng. Comm., 189: 1130 1154, 2002 Copyright # 2002 Taylor & Francis 0098-6445/02 $12.00 + .00 DOI: 10.1080/00986440290012555

CLASSIFICATION OF STATIC AND DYNAMIC BEHAVIOR IN CHEMOSTAT FOR PLASMID-BEARING, PLASMID-FREE MIXED RECOMBINANT CULTURES

ABDELHAMID AJBAR Department of Chemical Engineering, King Saud University, Riyadh, Saudi Arabia The singularity theory is combined with continuation techniques to classify the static and dynamic behavior in a chemostat involving the competition between plasmid-bearing and plasmid-free cell populations. The analysis of the static bifurcation allows the derivation of analytical conditions for the coexistence of the competing populations and for the safe operation of the bioreactor. The analysis of dynamic bifurcation, on the other hand, shows the ability of the model to predict the coexistence of the two populations in a state of stable limit cycle. Analytical conditions with respect to any growth kinetics are derived for the occurrence of Hopf points in the model. The combination of results of both static and dynamic bifurcation helps to construct a useful picture, in the multidimensional parameter space, of the different behavior predicted by the model. Keywords: Plasmid; Recombinant culture; Chemostat; Stability; Oscillations

INTRODUCTION The instability of the recombinant plasmid DNA has been the subject of a number of experimental and theoretical investigations (Imanaka and Aiba, 1981; Ollis and Chang, 1982; Nordstrom and Aagaard-Hansen, 1984; Lee and Bailey, 1984; Ryder and DiBiasio, 1984; Stephanopoulos Received 24 March 2000; in ®nal form 30 March 2001. Address correspondence to Abdelhamid Ajbar, Department of Chemical Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia. Fax: 966-1-467-8770. E-mail: [email protected] 1130

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and Lapidus, 1988; Macken et al., 1994; Hsu et al., 1994). A number of theoretical models have been proposed in the literature to describe the stability behavior in continuous cultures of mixed recombinant DNA. Using simple unstructured models, these studies (Ryder and DiBiasio, 1984; Stephanopoulos and Lapidus, 1988; Macken et al., 1994; Hsu et al., 1994) have shown that a fair understanding of the stability problem can be achieved through the analysis of the dynamics of the competition between plasmid-bearing and plasmid-free cell populations. Ryder and DiBiasio (1984), for instance, carried out a stability study for general growth kinetics based on linearized eigenvalue analysis. The authors also suggested an operational strategy involving feedback control to enhance plasmid stability in the chemostat. Stephanopoulos and Lapidus (1988) used index theory arguments to determine the steady state portraits based on the shape and mutual disposition of the speci®c growth rates, for both Monod and substrate inhibition kinetics. Hsu et al. (1994), on the other hand, carried out a global stability analysis of the chemostat for a class of noninhibitory growth kinetics. In this paper the problem of plasmid stability and strain reversion in recombinant cultures is revisited through a complete stability analysis of a plasmid-bearing, plasmid-free mixed culture growing in a chemostat. The stability analysis is carried out using elementary concepts of the singularity theory and continuation techniques. The singularity theory (Golubitsky and Schaeffer, 1985) can provide a general framework for classifying branching phenomena in which different kinds of multiplicity in the nonlinear model of the bioreactor can be expected. The theory has found several applications in the chemical engineering ®eld (Balakotaiah and Luss, 1982; D'anna et al., 1986; Alhumaizi and Aris, 1995). Recently, Ajbar et al. (Ajbar and Ibrahim, 1997; Ajbar and Alhumaizi, 2000; Ajbar, 2001) have successfully applied the theory to analyze the stability characteristics of the basic unstructured model of the continuous bioreactor (Ajbar and Ibrahim, 1997), to study global branching phenomena in pure and simple microbial competition (Ajbar and Alhumaizi, 2000), and to study microbial growth with wall attachment (Ajbar, 2001). The theory can provide a systematic classi®cation of the multidimensional parameter space into regions of different static and dynamic behavior. This classi®cation can have practical impacts on the operation of the bioreactor, since it can allow the determination of boundaries between safe and wash-out regions, and the delineation of regions of undesired oscillations or instability. Two objectives are speci®cally sought for this paper. The ®rst is to provide a uni®ed framework, using the singularity theory, for the analysis of static bifurcation induced in the bioreactor by the competing organisms. A practical picture of branching phenomena in the bioreactor, including different types of static coexistence, can be constructed in a systematic and quite useful manner. The different steady state portraits

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can also be used as a guide for selecting the operating conditions to enhance the stability and robustness of the culture. The relative simplicity of the unstructured model makes the analysis even more useful, since closed analytical results can be obtained to describe the different conditions for the coexistence of the competing cultures and for the safe operation of the chemostat. This also allows a useful analysis of the effect of the chemostat design and operating parameters on its different expected behaviors. The second objective of the paper is to study the dynamic bifurcation induced by the competing organisms. In the aforementioned studies, the issue of the existence of a periodic behavior in the chemostat was not addressed. In contrast to linear systems, it is known that structurally stable periodic solutions can exist for nonlinear systems. In this work, analytical results are derived for the occurrence of periodic behavior in the model with respect to any growth kinetics. The combination of results of both static and dynamic analysis helps to construct a complete picture of the different modes of behavior induced in the bioreactor by the competing populations. An unstructured model of the bioreactor with Andrews's kinetics (Andrews, 1968) is selected to describe substrate inhibition effects associated with the growth of the two organisms. The stability characteristics for the case of noninhibitory (Monod) growth model can be simply derived as a limiting case of the Andrews expressions and are therefore easily incorporated within the general treatment offered in this work by the singularity theory. The organization of the paper includes a presentation of the model of the chemostat. Static analysis is ®rst carried out, followed by a study of the dynamic bifurcation. For the sake of clarity, the mathematical details of the singularity theory are omitted from the paper. Most of the related materials about the theory can be found in the referenced textbook (Golubitsky and Schaeffer, 1985) or in the useful summaries (Alhumaizi and Aris, 1995; Ajbar and Ibrahim, 1997). PROCESS MODEL The unsteady state mass balances for the limiting substrate S, the plasmid-bearing X1, and plasmid-free X2 species are given in the following:   dS r1 X 1 r2 X 2 ˆ Sf S y ‡ y …1† dt Y1 Y2 y

dX1 ˆ dt

X1 ‡ …1

y

dX2 ˆ dt

X2 ‡ qyr1 X1 ‡ yr2 X2

q†yr1 X1

…2† …3†

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Yi (i ˆ 1,2) are the yield constants, q the probability that upon division a plasmid is lost or becomes modi®ed, and y is the reactor residence time. The two competing populations X1 and X2 grow on substrate S following Andrews's (Andrews, 1968) substrate inhibition expressions rj ˆ

mj S Kj ‡ S ‡ S2 =KIj

… j ˆ 1; 2†

…4†

where KIj … j ˆ 1; 2† are the inhibition constants. The mass balances are suitably rendered dimensionless using the variables shown in Table I. The dimensionless mass balance equations for the different species are, therefore, dS  ˆ Sf dt

S

 y… r1 X1 ‡ r2 X2 †

…5†

q† y r1 X1

…6†

 y r2 † ‡ qZ y r1 X1

…7†

dX1 ˆ dt

X1 ‡ …1

dX2 ˆ dt

X2 …1

The dimensionless expressions of the speci®c growth rates are given, on the other hand, by

Table I Dimensionless Variables Used in the Model Parameter

De®nition

y

m1 y

S

S K1

t

t y

X1

X1 Y1 K1

X2

X2 Y2 K1

a

K2 K1

g1

K1 KI1

g2

K1 KI2

Z

Y1 Y2

f

m2 m1

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 r1 ˆ

A. AJBAR

S  1 ‡ S ‡ g1 S2

and

r2 ˆ

fS  a ‡ S ‡ g2 S2

…8†

The noninhibitory (Monod) growth models are obtained as a limiting case of Equation (8) as the dimensionless inhibition constants g1 and g2 take on small values, i.e., gj ! 0; j ˆ 1; 2: STATIC BIFURCATION The steady state equations are obtained by setting the lefthand sides of Equations (5 7) to zero. Equations (6 7), in particular, would yield q† y r1 † ˆ 0

X1 … 1 ‡ …1

…9†

and Zq y r1 X1 X2 ˆ 1  y r2

…10†

Inspection of these equations reveals that the following three steady states are possible:  X1 ˆ 0; X2 ˆ 0, and S ˆ Sf . This corresponds to the total wash-out condition.  X1 ˆ 0 and X2 > 0, which corresponds to wash-out of plasmid-bearing cell population.  X1 > 0 and X2 > 0, which corresponds to the coexistence of the two populations. We are interested in the way the steady state S depends on the positive system parameters. The residence time …y† is the easiest parameter to vary and is selected to be the bifurcation parameter. The continuity  diagram …S; y† always has as a solution the total wash-out line …S ˆ Sf †, in addition to the curves corresponding to the plasmid-bearing wash-out and=or that corresponding to the coexistence of the two populations. In the following we analyze, using elementary principles of the singularity theory, the conditions for the existence of these solutions. The wash-out of plasmid-bearing cell population corresponds to  :ˆ r2 F…S†

1 and X2 ˆ Sf  y

S

…11†

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Recasting the expression of  r2 (Equation (8)), the ®rst term of Equation (11) de®nes a quadratic equation,  :ˆ aS2 ‡ bS ‡ c ˆ 0 F…S†

…12†

with a ˆ g2 ;

bˆ1

 yf;

cˆa

…13†

For a quadratic equation such as Equation (13), the singularity theory de®nes a simple bifurcation (codimension 0) de®ned by the following conditions: F ˆ 0;

FS ˆ 0;

and

FSS 6ˆ 0

…14†

The equation FS ˆ 0 de®nes the condition for the existence of a static  limit point in the continuity diagram …S; y†. Taking the derivative of F (Equation (11)) yields FS ˆ r20 ˆ 0

…15†

Noting that r20 ˆ

f…a g2 S2 † …a ‡ S ‡ g2 S2 †2

…16†

q it can be seen that FS vanishes at S ˆ ga . Since S is always bounded by 2 the feed condition Sf , a necessary condition, therefore, for the existence of a static limit point is that a g2  2 Sf

…17†

Should this condition be satis®ed, the equation Fs ˆ 0 (Equation (12)) de®nes one static limit point S ˆ

b 2a

…18†

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The condition FSS ˆ 2a ˆ 2g1 , on the other hand, vanishes only when the inhibition-constant g1 is zero, i.e., Monod kinetics. Substituting Equation (18) into Equation (12) yields b2 ˆ 4ac

…19†

By recasting the expressions of a; b, and c from Equation (13), the condition of Equation (19) de®nes the residence time yc1 corresponding to the static limit point, 1 p  yc1 ˆ …1  2 g2 a† f

…20†

The coexistence of the two populations, on the other hand, is possible when the following condition (Equation (9)) is satis®ed:  :ˆ r1 F…S†

1 …1

q† y

ˆ0

…21†

The concentrations of the competing populations are found using Equations (5) and (10), which yield …Sf X1 ˆ

 S†‰…1 q† r1 1 Zq r1 1 q r2

 r2 Š

…22†

and X2 ˆ

ZqX1  r1 …1 q† r1  r2

…23†

The coexistence is meaningful, i.e., X2 > 0, provided that …1

q† r1

 r2 > 0

…24†

Should this condition be satis®ed, then Equation (22) also yields  positive values for X1 since the term r1 11 Zq q r2 will also be positive. The condition of Equation (21) also de®nes a quadratic equation, similar to Equation (12) with a ˆ g1 ;

bˆ1

…1

q† y;

cˆ1

…25†

Similar to Equation (17), the necessary condition for the occurrence of a static limit point in the continuity diagram is that 1 g1  2 Sf

…26†

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The static limit point occurs at the residence time p …1  2 g1 †  yc2 ˆ 1 q

…27†

The condition for the meaningful coexistence (Equation (24)) is equivalent to aS2 ‡ bS ‡ c > 0

…28†

with a ˆ …1

q†g2

fg1 ; b ˆ …1

q†

f; c ˆ a…1

q†

f

…29†

The discriminate D ˆ b2

4ac > 0

…30†

with either one or both roots Sj … j ˆ 1; 2†, satisfying 0 < Sj ˆ

p b D  Sf … j ˆ 1; 2† 2a

…31†

de®nes the boundaries for the coexistence of the competing populations. Moreover, the two curves …X1 ˆ 0; X2 > 0† and …X1 > 0; X2 > 0†  may cross in the continuity diagram …S; y†. The crossing occurs when the ®rst term of Equation (11) and the condition of Equation (21) are satis®ed. This corresponds to the roots of the coexistence condition (Equation (28)). The corresponding residence time can be obtained from Equation (21). The relative position of the two curves in the continuity diagram depends on the relative location of the two static limit points. The condition  yc1 ˆ  yc2 (Equations (20) and (27)) is equivalent to  p 1  2 g1 1 f g2 ˆ 4a 1 q

1

2

…32†

This relation de®nes a boundary separating two modes of behavior, as will be seen in a later section. A ®nal note concerns the safe operation of the chemostat. The points of crossing of each of the two curves with the total wash-out line …S ˆ Sf †

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de®ne conditions for the safe operation of the chemostat. These conditions are obtained by setting S ˆ Sf in Equations (11) and (21), respectively, leading to  yw1 ˆ

1 ‡ Sf ‡ g1 S2f …1 q†Sf

…33†

a ‡ Sf ‡ g2 S2f fSf

…34†

and  yw2 ˆ

Elementary principles of the singularity theory, therefore, have allowed the derivation of analytical conditions de®ning the complete static behavior of the chemostat. Before we present examples of these branch sets we investigate in the following section the dynamic bifurcation of the model, that is, the possibility of the competing populations coexisting in a state of limit cycle. EXISTENCE OF PERIODIC SOLUTIONS The existence of periodic solutions, i.e., Hopf points, in the model is associated with a change in the equilibrium of singular points when a single pair of eigenvalues of the linearized system crosses the imaginary axis. The three-dimensional system has a Hopf bifurcation point when the Jacobean matrix J has pure imaginary eigenvalues. The eigenvalues l of the Jacobean matrix are the solutions of the characteristic matrix equation l3 ‡ T1 l2

T2 l ‡ T3 ˆ 0

…35†

where T1 ; T2 , and T3 are the three invariants of J, T1 ˆ j11 ‡ j22 ‡ j33  T2 ˆ det

j11 J21

T3 ˆ det…J†

j12 j22

…36† 

 ‡ det

j22 J32

j23 j33



 ‡ det

j11 J31

j13 j33

 …37† …38†

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The j11 ; j12 ; . . . are the elements of J. The conditions of Hopf bifurcation in terms of the coef®cients T1 ; T2 , and T3 can be derived by setting l ˆ iw into Equation (35) to yield F1 :ˆ T1 T2

T3 ˆ 0

…39†

T2 > 0

…40†

The Jacobean terms are given explicity by taking the derivatives of Equations (5 7), yielding j11 ˆ

 y… r10 X1 ‡ r20 X2 †;

1

j21 ˆ …1

q† y r10 X1 ;

j22 ˆ

j31 ˆ qZ y r10 X1 ‡  y r20 X2 ;

j12 ˆ

 y r1 ;

j13 ˆ

q† y r10 ;

1 ‡ …1

j32 ˆ qZ y r1 ;

j33 ˆ

y r2

j23 ˆ 0 1 ‡ y r2

…41†

The ri0 …i ˆ 1; 2† are the ®rst derivatives of ri . Similar to the static case, a number of singularities can be de®ned for the dynamic behavior. The simplest interactions between a Hopf point and a static limit point occur when the imaginary part of the complex conjugate eigenvalue pair goes to zero. This degeneracy, called the F1 degeneracy (Golubitsky and Schaeffer, 1985), is de®ned by solving the steady state equation (Equations (11) and (21)), the condition T2 ˆ 0, (Equation (37)), and the static limit point condition T3 ˆ 0 (Equation (38)). Evaluating these conditions using the software Mathematica (Wolfram Research, Inc.) for the important case of the coexistence of the two cultures (i.e., 1 ‡ …1 q† y r1 ˆ 0) yields T3 ˆ …1

y2 X1 ‰ 1 ‡  q† r1 r10  y r2 …1

Zq†Š

…42†

Barring the trivial case of r1 ˆ 0; i.e., S ˆ 0, it can be seen that T3 vanishes when either r01 ˆ 0

…43†

or 1‡ y r2 …1

Zq† ˆ 0

…44†

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The ®rst condition r01 ˆ 0 is equivalent to the occurrence of a static limit point. Substituting this condition in the term T2 yields T2 ˆ 1

 y ‡ r02  yX2 r2 

…45†

At the F1 singularity, this terms vanishes, which yields y 1 r2  X2 ˆ  0 y r2

…46†

The concentration of the plasmid-bearing cell population is obtained from the steady state equation (Equation (23)) …1 X1 ˆ

 y r2 †X2  Zqy r1

…47†

The residence time at which the F1 singularity occurs is obtained from the steady state equation (Equation (21))  yˆ

1 r1 …1 q†

…48†

Substituting these conditions in the steady state form of Equation (5) yields the following general condition for the occurrence of the F1 singularity: Sf

‰…1 SH ‡

q† r1

 r2 Š‰…1 q† r1 …1 Zq…1 q† r1 r02

Zq†r2 Š

ˆ0

…49†

with SH satisfying the condition r01 …SH † ˆ 0. Equation (49) represents, therefore, general conditions with respect to any growth kinetics for the occurrence of the F1 singularity. When crossing the curve(s) de®ning this singularity, the number of Hopf points in the continuity diagram increases=decreases by one. The condition of Equation (49) is reduced for the case of equal yield (Z ˆ 1) to the simpler form Sf

‰…1 SH ‡

q† r1  r2 Š‰ r1 0 q r1  r2

 r2 Š

ˆ0

…50†

It can be noted that these results cannot hold when the plasmid-bearing organism grows following Monod kinetics, i.e., g1 ˆ 0, since r01

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 2 and therefore cannot (Equation (8)) is reduced in this case to 1=…1 ‡ S† vanish. Next we apply these results for the selected Andrews' kinetics (Equation (8)). For this case, the term  r01 vanishes at SH ˆ p1 g . Sub1

stituting in Equation (49) yields the following relation involving the chemostat kinetic and operating parameters for the occurrence of the F1 singularity: 1 Sf ˆ p g1 ‡

g2 g2 p1 p1 p1 ‰…1 q†…a‡ p1 g1 ‡ g1 † f…2‡ g1 †Š‰…1 q†…a‡ g1 ‡ g1 † …1 Zq†f…2‡ g1 †Š   p Zq…1 q†f…a gg2 †…2 g1 ‡1† 1

…51† The other possibility for T2 to vanish (Equation (44)) is not physically realistic. Using the steady state equation (Equation (21)), it can be seen that the term …1 q† r1  r2 is reduced to Zq r2 , which is always negative and hence yields physically unrealistic values for X2 (Equation (23)). The above analytical results for either static or dynamic bifurcation consist of simple algebraic equations that can be readily solved for any set of numerical parameters. Next we present some numerical simulations for the different branching phenomena. The following parameters are selected (Imanaka and Aiba, 1981; Ollis and Chang, 1982; Ryder and Di Biasio, 1984): a ˆ 0:015, f ˆ 0:5, q ˆ 0:15, Z ˆ 1, and Sf ˆ 3. This corresponds to b > 0 and c < 0 in the coexistence condition (Equations (28) and (29)). Figure 1a, also enlarged in Figure 1b, shows the complete static and dynamic branch set in the parameter space (g1 ; g2 ). A total of seven different regions can be found. The ®rst solid line (from the right) separating region (a) and the smaller region (b) (enlarged in Figure 1b) corresponds to the discriminate (D) of the coexistence condition being zero (Equation (30)). In region (a) the discriminate is negative. Since the term c (Equation (29)) is also negative, this implies that the coexistence of the two populations is not feasible. This situation corresponds to the case where the plasmid-free organism grows always faster than the plasmid-bearing organism, as shown schematically in Figure 2a. The continuity diagram for this region is shown (Figure 2b), for instance, for …g1 ; g2 † ˆ …0:16; 0:016†. The continuity diagram, generated by the software AUTO (Doedel and Kernevez, 1986), is characterized by the presence of the plasmid-bearing wash-out curve in addition to the total wash-out line. The stability of plasmid-bearing wash-out is satis®ed for conditions above the static limit point (LP) de®ned by Equation (20).

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Figure 1. Static and dynamic branch sets (a); Enlargement of the branch set (b).

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Figure 2. Chemostat behavior in region (a) of Figure 1. (a) Growth rates: solid …1 q† r1 ; dash  r2 . (b) Corresponding continuity diagram: solid Ð stable branch; dash Ð unstable; ®lled circle Ð static limit point; empty circle Ð bifurcation point. (1) indicates the curve of plasmid-bearing wash-out.

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For conditions below LP a total wash-out occurs, while for conditions between (LP) and the point of crossing (BR) of the two curves, de®ned by Equation (34), two stable branches coexist. The system may also settle on the total wash-out line depending on start-up conditions. When crossing from region (a) to region (b) (Figure 1b), the discriminate of the coexistence condition becomes positive and two real roots exist. The smaller region (b) corresponds to the situation where the roots S1 and S2 satisfy 0 < S2 < S1 < Sf . The corresponding growth rates are shown in Figure 3a. The plasmid-bearing cell population grows faster than the plasmid-free in the region between S2 and S1 . The coexistence of the two competing populations is unstable in this region. The corresponding continuity diagram is shown in Figure 3b for (g1 ; g2 ) ˆ (0.16, 0.0204). In addition to the plasmid-bearing wash-out curve (curve (1)), an unstable coexistence curve (curve (2)) can be seen in the enlargement of the ®gure (Figure 3c). The bifurcation points BR1 and BR2 indicate the limits of the coexistence region that also correspond to the roots S1 and S2 . When moving to region (c) (Figure 1) one of the roots of the coexistence condition becomes larger than Sf . The situation, shown in Figure 4a, corresponds to 0 < S2 < Sf < S1 . In this case the plasmidbearing cell grows faster than the plasmid-free cell for S > S2 . The corresponding continuity diagram is shown in Figure 4b for (g1 ; g2 ) ˆ (0.16, 0.023). Both the plasmid-bearing wash-out (curve (1)) and coexistence curve (curve (2)) are characterized by the presence of static limit points de®ned, respectively, by Equation (20) and Equation (27). The coexistence is unstable between the crossing of the two curves (root S2 ) and the crossing with the wash-out line de®ned by Equation (33). The safe operation of the bioreactor is possible for conditions above LP1 de®ned by Equation (20), although bistability occurs between LP1 and BR1 . The line separating regions (c) and (d) (Figure 1) is the F1 curve (Equation (51)). When crossing this line a Hopf point necessarily appears in the continuity diagram. Region (d) is similar to region (c) as far as the crossing of the growth rates is concerned. However, because of the existence of a Hopf point the continuity diagram is dramatically different. Figure 5a shows an example of the behavior in this region for (g1 ; g2 ) ˆ (0.16, 0.04). It can be seen that besides the static limit points the continuity diagram is characterized by the presence of a Hopf point (HB) along the coexistence curve. The coexistence is stable between the static limit point LP2 (Equation (27)) and the Hopf point (HB). The enlargement of the continuity diagram (Figure 5b) shows that stable periodic branches emanate from the Hopf point and terminate homoclinically as they collide with the static branch. Periodic behavior is therefore expected in the region around the Hopf point.

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Figure 3. Chemostat behavior in region (b) of Figure 1b. (a) Growth rates: solid …1 q† r1 ; dash r2 . (b) Corresponding continuity diagram. (c) Enlargement of the continuity diagram: solid Ð stable branch; dash Ð unstable; ®lled circle Ð static limit point; empty circle Ð bifurcation point. (1) indicates the curve of plasmid-bearing wash-out; (2) curve of coexistence.

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Figure 4. Chemostat behavior in region (c) of Figure 1. (a) Growth rates: solid …1 q†r1 ; dash  r2 . (b) Corresponding continuity diagram: solid Ð stable branch; dash Ð unstable; ®lled circle Ð static limit point; empty circle Ð bifurcation point. (1) indicates the curve of plasmid-bearing wash-out line; (2) curve of coexistence.

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Figure 5. Chemostat behavior in region (d) of Figure 1. (a) Continuity diagram. (b) Enlargement of the continuity diagram showing stable periodic branches emanating from the HB point: solid Ð stable branch; dash Ð unstable; ®lled circle Ð static limit point; empty circle Ð bifurcation point; square Ð HB point. (1) indicates the curve of plasmid-bearing wash-out line; (2) curve of coexistence.

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The behavior of the system in this region is quite involved because of the existence of more than one stable attractor in each domain. For conditions below the static limit point LP1 , total wash-out occurs. For conditions between the static limit points LP1 and LP2 , the plasmidbearing wash-out curve coexists with the total wash-out line. Different start-up conditions may lead to either plasmid-bearing wash-out (curve (1)), or to total wash-out. For conditions between the static limit point LP2 and the Hopf point three stable branches coexist. Different start-up conditions may lead to a stable coexistence, to plasmid-bearing wash-out, or to a total wash-out. For the region around the Hopf point the system may settle on a plasmid-bearing wash-out curve or may exhibit stable sustained oscillations. An example of the limit cycle is shown in Figure 6  X1 ; X2 † ˆ …2:1; 0:1; 0:81†. The line for  yc ˆ 2:134 and initial conditions …S; separating region (d) and (e) (Figure 1) is de®ned by Equation (32). When crossing this line the relative location of the two curves changes in the continuity diagram but the relative location of the growth rates is unchanged from Figure 4a. Figure 7a shows an example of this situation for …g1 ; g2 † ˆ …0:16; 0:08†. It can be seen that the relative location of the static limit points LP1 and LP2 of the two curves has changed, compared to region (d). For conditions below LP2 (Equation (27)) total wash-out occurs. Moreover, the relative location of the two curves adds a new aspect to the behavior of the system. Unlike region (d), where the coexistence was dependent on start-up conditions, it can be seen in Figure 7b that a stable coexistence of the two competing organisms is the sole attractor to the system for the whole region between LP2 (Equation (27)) and LP1 (Equation (20)). All startup conditions would lead to a stable coexistence. Moreover, for the region between LP1 and the Hopf point, the system may settle on a stable coexistence or on plasmid-bearing wash-out, while for the region around the Hopf point, stable oscillations are expected. When crossing the line (Equation (26)) de®ning the static limit point (LP) (Equation (26)) and into region (f) of Figure 1, the static limit point disappears from the coexistence curve. Figure 8(a c) shows an example of the behavior in this region for (g1 ; g2 ) ˆ (0.05, 0.08). Wash-out conditions occur for conditions below BR1 de®ned by Equation (33). The absence of the static limit point also has its effects on the dynamic behavior of the competing cultures. A stable coexistence is possible, for all start-up conditions, for the whole region extending from the BR1 to the Hopf point (HB). Moreover, sustained oscillations are also expected around the Hopf point. Finally, in the quadrant (g) of Figure 1a, both lines (Equations (17) and (26)) de®ning the static limit points are crossed. The two static limit points as well as the Hopf point disappear from the two curves corresponding to plasmid-bearing wash-out and static coexistence. Figure 9(a b) shows the behavior in this region for (g1 ; g2 ) ˆ (0.05, 0.001). A monotonic behavior

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Figure 6. Examples of periodic behavior in region (d) (Figure 5b) for y ˆ 2:134. Sustained  X1 ; X2 † ˆ …2:1; 0:1; 0:81†. oscillations are obtained with the initial conditions …S;

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Figure 7. Chemostat behavior in region (e) of Figure 1. (a) Continuity diagram. (b) Enlargement of the continuity diagram showing stable periodic branches emanating from the HB point: solid Ð stable branch; dash Ð unstable; ®lled circle Ð static limit point; empty circle Ð bifurcation point; square Ð HB point. (1) indicates the curve of plasmid-bearing wash-out line; (2) curve of coexistence.

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Figure 8. Chemostat behavior in region (f) of Figure 1. (a) Growth rates: solid …1 q† r1 ; dash  r2 . (b) Continuity diagram. (c) Enlargement of the continuity diagram showing stable periodic branches emanating from the HB point: solid Ð stable branch; dash Ð unstable; ®lled circle Ð static limit point; empty circle Ð bifurcation point; square Ð HB point. (1) indicates the curve of plasmid-bearing wash-out line; (2) curve of coexistence.

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A. AJBAR

Figure 9. Chemostat behavior in region (g) of Figure 1a. (a) Growth rates: solid …1 q†r1 ; dash r2 . (b) Continuity diagram: solid Ð stable branch; dash Ð unstable; ®lled circle Ð static limit point; empty circle Ð bifurcation point. (1) indicates the curve of plasmid-bearing wash-out line. (2) curve of coexistence.

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similar to Monod kinetics prevails. It can be seen that the coexistence is stable for conditions between BR1 (Equation (33)) and the crossing of the two curves. CONCLUSIONS A static and dynamic analysis of a chemostat for the competition between plasmid-bearing and plasmid-free cell populations was carried out using elementary concepts of the singularity theory and continuation techniques. Using a simple unstructured model, analytical conditions were derived for the static coexistence of the competing populations and for the safe operation of the chemostat, for both substrate inhibition and Monod kinetics. The interesting problem of coexistence of the competing populations in a state of limit cycle was solved. General analytical conditions for the existence of Hopf points in the model with respect to any growth kinetics were derived. The combination of results for both the static and dynamic analysis allowed the systematic construction of branch sets in the multidimensional parameter space where different behavior was delineated. Numerical simulations carried out for some values of model parameters allowed the identi®cation of domains where the coexistence of the competing populations is the sole outcome of the competition as well as regions where bistability exists and for which ¯uctuations in feed conditions can break the static coexistence and push the process either to plasmid bearing wash-out or to total wash-out conditions. The existence of a periodic behavior added interesting features to the outcome of the competition. It was shown that for relatively large parameter sets, the model predicts the coexistence of the plasmid-free and plasmid-bearing populations in a state of stable limit cycle. However, the oscillatory behavior was found to always coexist with static branches. Fluctuations in feed conditions may therefore annihilate the oscillatory behavior and push the process to plasmid-bearing wash-out. The dynamic analysis has also shown that the existence of a Hopf point, as result of the F1 degeneracy, is necessarily limited to nonmonotonic growth rates. However, other dynamic singularities should not be excluded. One interesting singularity concerns the potential existence of stable periodic branches connecting two Hopf points. The conditions on growth kinetics for the existence of this singularity deserve further investigation. NOTATION KIj Kj

substrate inhibition constants in speci®c growth rates rj ( j=1,2) (mass volume 1 ) constant in the speci®c growth rates rj ( j ˆ 1,2) (volume mass 1 )

1154 rj S t X1 X2 Yj

A. AJBAR speci®c growth rates of population j (j ˆ 1,2) (time 1 ) concentration of substrate S (mass volume 1 ) time concentration of plasmid-bearing cell population (mass volume 1 ) concentration of plasmid-free cell population (mass volume 1 ) yield coecient of species j (j ˆ 1,2) (massXj massS 1 )

Greek Letters a dimensionless constant in r2 gj dimensionless inhibition constant in rj (j ˆ 1,2) 1 Z ratio of yield coecient …Y Y2 † y dimensionless residence time f dimensionless constant in r2 Subscripts f feed …:† dimensionless value of (.) Abbreviations BR bifurcation point HB Hopf bifurcation point LP static limit point

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